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Clifford spectrum of three 2 by 2 matrices

Alexander Cerjan, Vasile Lauric, Terry A. Loring

TL;DR

This work proves that the Clifford spectrum for any triple of $2\times2$ Hermitian matrices is nonempty by deriving an explicit quartic determinant $D(x)$ for the localizer $L_{x}(A)$ after reducing to a Pauli-basis representation. The authors compute $D(x)$, show its symmetry, and use the condition $D(x)\le0$ to guarantee singularity of the localizer for some $x$, thereby ensuring nonemptiness. They further reinterpret the zero set $D(x)=0$ geometrically as intersections of Cassini 2d-ovals with a hyperboloid in $\mathbb{R}^3$, leading to a rich structure where the Clifford spectrum can be a single point, two points, or two disjoint components, depending on parameters. The results highlight a concrete, structure-rich picture of the Clifford spectrum that contrasts with other noncommutative spectra and connects to topological physics via the spectral localizer, with explicit examples and a detailed determinant calculation supporting the conclusions.

Abstract

We prove that the Clifford spectrum associated to three 2 by 2 matrices is nonempty. The structure of Clifford is described in terms "moving" level curves. We discuss some implication of a conjecture formulated for arbitrary size n by n of three matrices and give an example in the case of three self-adjoint operators in the infinite dimensional Hilbert space.

Clifford spectrum of three 2 by 2 matrices

TL;DR

This work proves that the Clifford spectrum for any triple of Hermitian matrices is nonempty by deriving an explicit quartic determinant for the localizer after reducing to a Pauli-basis representation. The authors compute , show its symmetry, and use the condition to guarantee singularity of the localizer for some , thereby ensuring nonemptiness. They further reinterpret the zero set geometrically as intersections of Cassini 2d-ovals with a hyperboloid in , leading to a rich structure where the Clifford spectrum can be a single point, two points, or two disjoint components, depending on parameters. The results highlight a concrete, structure-rich picture of the Clifford spectrum that contrasts with other noncommutative spectra and connects to topological physics via the spectral localizer, with explicit examples and a detailed determinant calculation supporting the conclusions.

Abstract

We prove that the Clifford spectrum associated to three 2 by 2 matrices is nonempty. The structure of Clifford is described in terms "moving" level curves. We discuss some implication of a conjecture formulated for arbitrary size n by n of three matrices and give an example in the case of three self-adjoint operators in the infinite dimensional Hilbert space.
Paper Structure (4 sections, 4 theorems, 86 equations, 1 figure)

This paper contains 4 sections, 4 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Three two-by-two hermitian matrices.
  3. Proof of Proposition 2.1
  4. Geometrical structure of the Clifford spectrum

Key Result

Proposition 2.1

where $\alpha_2=a_{22}+i\, a_{21}$ and $\alpha_3=a_{31}+i\, a_{32}.$

Figures (1)

  • Figure 4.1: Surfaces corresponding to the Clifford spectrum for the matrix triplets given in Example \ref{['ex:1']} (a), Example \ref{['ex:2']} (b), and Example \ref{['ex:3']} (c).

Theorems & Definitions (10)

  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Theorem 2.4
  • proof
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 4.5